It is interesting to investigate congruences of sums \(T_{m,k}(n)=\sum _{x=1}^{[n/m]}\frac{\chi _n(x)}{x^k}\)  involving the Euler quotient, generalized Bernoulli numbers
and Bernoulli polynomials modulo integer powers. For example, in 1938, Lehmer (see [1]}) established the famous congruence
\(\sum _{i=1}^{(p-1) / 2} \frac{1}{i} \equiv -2 q_2(p)+p q_2^2(p) \quad \left(\bmod p^2\right)\qquad \mathrm {(1.1)}\) 
